ASSIGNMENT PROBLEM WITH GENERALIZED INTERVAL ARITHMETIC · ASSIGNMENT PROBLEM WITH GENERALIZED INTERVAL ARITHMETIC . G. Ramesh. and. K. Ganesan. Abstract - Linear assignment problems

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 81

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ASSIGNMENT PROBLEM WITH

GENERALIZED INTERVAL ARITHMETIC

G. Ramesh and K. Ganesan

Abstract - Linear assignment problems are very well known linear programming problems of assignment problem. In a

ground reality the entries of the cost matrix are not always crisp. In many application this parameters are uncertain and this

uncertain parameters are represented by interval. In this contribution we propose a new computational procedure to solve

assignment problem with generalized interval arithmetic interval Hungarian method.

Index Terms— Interval Numbers, generalized interval arithmetic, Interval Linear Programming, Ranking

—————————— ——————————

1. INTRODUCTION In practical field we are sometime faced with type of problem which consists of jobs to machines, drivers to trucks, men to offices etc. in which the assignees possess varying degree of efficiency, called as cost or effectiveness. The basic assumption of this type of problem is that one person can perform one job at a time. An assignment plan is optimal if it minimizes the total cost or maximizes the profit. This type of linear assignment problems can be solved by the very well-known Hungarian method which was derived by the two mathematician Kuhn, H. W [14]. Authors are proposed different methods to handle different types of assignment problems. In this context, Albrecher [1, 10] introduced an asymptotic behaviour of bottleneck problems and Aldous [3] studied asymptotes in the random assignment problem. Alvis and Lai [4] coined out the probabilistic analysis of a heuristic for the assignment problem .Quadratic assignment was analyzed by many authors in different approaches [4, 5, 7, 8]. Burkard [6] proposed time-slot assignment for TDMA systems .Burkard et al. [9] studied an algebraic approach to assignment problems. Pardalos and Pitsoulis [22] developed some works on nonlinear assignment problems. Asymptotic properties of the random assignment problem have been studied by Olin [21]. Nair Proved of the Parisi and Coppersmith-Sorkin conjectures in the random assignment problem [20]. Fitness landscape analysis and memetic algorithms for the quadratic assignment problem are described by Merz and Freisleben [18]. Optimal Permutations and Bottleneck quadratic assignment problems and the bandwidth problem were

analyzed by Li et al. and Kellerer et al. [ 20]. Krokhmal et al. [15] described asymptotic behaviour of the expected optimal value of the multidimensional assignment problem. In the realistic problems costs are not always in a crisp form, sometime these parameters are uncertain which are represented by intervals. Hence we need the help of interval analysis for handling this type of data. In this paper a general interval linear assignment problem is taken into consideration with basis assumption that one person can perform one job at a time. Here the new method has been proposed to handle such type of problem. We solved one example problem using this proposed method. Corresponding results are computed and has been reported here. The rest of the paper is organized as follows: in the next section review on interval arithmetic are highlighted. In section 3 the detail of proposed interval Hungarian method are presented.In section 4 example problems are solved and results are analyzed .Finally conclusions are drawn in section 5.

2. PRELIMINARIES The aim of this section is to present some notations,

notions and results which are of useful in our further

consideration.

Let 1 2 1 2a=[a , a ] = {x :a x a ,x R}. If 1 2a= a = a = a,

then a=[a, a] = a is a real number (or a degenerate

interval). Let 1 2 1 2 1 2IR = {a = [a , a ] : a a and a , a R} be the

set of all proper intervals and

1 2 1 2 1 2IR = {a = [a , a ] : a > a and a , a R} be the set of all

improper intervals on the real line R. We shall use the

terms”interval” and”interval number” interchangeably.

The mid-point and width (or half-width) of an interval

number 1 2a [a , a ] are defined as 1 2a + am(a) =

2

and

2 1a - aw(a) = .

2

The interval number a can also be

G. Ramesh

Department of Mathematics,

Faculty of Engineering and Technology, SRM University, Kattankulathur, Chennai - 603203, India Email: [email protected]

K. Ganesan Department of Mathematics, Faculty of Engineering and Technology, SRM University, Kattankulathur, Chennai - 603203, India Email:[email protected], [email protected]

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expressed in terms of its midpoint and width as

1 2a [a ,a ] m(a),w(a) .

2.1. A New Interval Arithmetic

Ming Ma et al. Error! Reference source not found.

have proposed a new fuzzy arithmetic based upon

both location index and fuzziness index function .

The location index number is taken in the ordinary

arithmetic, whereas the fuzziness index functions are

considered to follow the lattice rule which are the least

upper bound and greatest lower bound in the lattice L.

That is for a,b L we define a b = max{a,b} and

a b = min{a,b}.

For any two intervals 1 2a = [a , a ], 1 2b = [b , b ] IR and

for * +, -, ·, ÷ , the arithmetic operations on a and b

are defined as:

1 2 1 2a * b = [a , a ]*[b , b ] = m(a), w(a) m(b), w(b)

m(a) m(b), max w(a), w(b) .

In particular

a + b = m(a), w(a) m(b), w(b)


a - b = m(a), w(a) m(b), w(b)


a b = m(a), w(a) m(b), w(

(i).Addition :

(ii). Subtraction :

(iii). Multiplication :

b)


a b m(a), w(a) m(b), w(b)

m(a) m(b), max{w(a), w(b)} ,

provided m(b) 0.

(iv). Division :

2.2. Ranking of Interval Numbers

Sengupta and Pal [2] proposed a simple and efficient

index for comparing any two intervals on IR through

decision maker’s satisfaction.

Definition 2.2.1. Let be an extended order relation

between the interval numbers 1 2a = [a , a ], 1 2b = [b , b ] in

IR, then for we construct a premise

(a b)° which implies that a is inferior to b (or b is

superior to a ).

An acceptability function A : IR IR [0, ) is defined

as:

A may be interpreted as the grade of acceptability of the

“the first interval number to be inferior to the second

interval number”. For any two interval numbers a and

b in IR either A(a b) 0 (or) A(b a) 0 (or)

A(a b) = 0 (or) A(b a) = 0 (or) A(a b) + A(b a) = 0 .

If A(a b) = 0 and A(b a) = 0, then we say that the interval

numbers a and b are equivalent (non-inferior to each

other) and we denote it by a b. Also if A(a b) 0,

then a b and if A(b a) 0, then b a.

3. MAIN RESULTS

3.1. General Interval Assignment Problem

Let there are n jobs and n persons are available with

different skills. If the cost of doing jth work by ith person is

cij .Now the problem is which work is to be assigned to

whom so that the cost of completion of work will be

minimum. Mathematically, we can express the problem

as follows:

Minimize Z (cost) = n n

ij iji 1 j 1

c x ; i 1,2,...n; j 1,2,...n

where ijx =

th th

th

th

1; if i person is assigned j work

0; if i person is not assigned the

j work with the restrictions

n

iji 1

x 1; j 1,2,...n.,

(1)

i.e., ith person will do only one work.

n

ijj 1

x 1; i 1,2,...n.,

(2)

i.e., jth work will be done only by one person.

3.2. Interval Hungarian Method

In this section an algorithm to solve assignment problem

with generalized interval arithmetic using Hungarian

method:

Step 1 Find out the mid values of each interval in the cost

matrix.

Step 2 Subtract the interval which have smallest mid

value in each row from all the entries of its row.

m(b) - m(a)(a,b) A(a b) , where w(b) w(a) 0.A

w(b) w(a)

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Step 3 Subtract the interval which have smallest mid

value from those columns which have no intervals

contain zero from all the entries of its column.

Step 4 Draw lines through appropriate rows and

columns so that all the intervals contain zero of the cost

matrix are covered and the minimum number of such

lines is used.

Step 5 Test for optimality (i) If the minimum number of

covering lines is equal to the order of the cost matrix, then

optimality is reached. (ii) If the minimum number of

covering lines is less than the order of the matrix, then go

to step 6.

Step 6 Determine the smallest mid value of the intervals

which are not covered by any lines .Subtract this entry

from all un-crossed element elements and add it to the

crossing having an interval contain zero. Then go to step

4.

3.3. Tabular form of the Problem The cost matrix of the interval assignment problem is given in the table below: Table 1. Cost matrix of the interval assignment problem

Persons Jobs

1 2 3 …..j …..n

1 11C 12C 13C ….. 1jC ….. 1nC

2 21C 22C 23C ….. 2 jC ….. 2nC

.

.

. i

.

.

.

i1C

.

.

.

i2C

.

.

.

i3C

.

.

.

ijC

.

.

.

inC

.

.

. n

.

.

.

n1C

.

.

.

n2C

.

.

.

n3C

……. ……. …….

…… njC

…… …… ……

…… nnC

4. NUMERICAL EXAMPLES Example 4.1

Let us consider an assignment problem discussed by

Sarangam Majumda [23]. The assignment cost of

assigning any operator to any one machine is given in the

following table.

Table 2. Cost matrix with crisp entries

I II III IV

A 10 5 13 15 B 3 9 18 3 C 10 7 3 2

D 5 11 9 7

By applying Hungarian method, Sarangam Majumdar

got an optimal assignment as A, B, C, D machines are

assign to II, IV, III, I operators respectively and minimum

assignment cost is 16.

They converted this crisp assignment problem as an

interval assignment problem as given below. Now the

cost matrix of the interval assignment problem is

Table 3. Cost matrix with interval entries

I II III IV

A [9,11] [4,6] [12,14] [14,16]

B [2,4] [8,10] [17,19] [2,4]

C [9,11] [6,8] [2,4] [1,3]

D [4,6] [10,12] [8,10] [6,8]

Applying their interval Hungarian method, Sarangam

Majumdar obtained the optimal assignment as A, B, C, D

machines are assign to II, I, III, IV operators respectively

and the optimum assignment cost as [14, 22]. After stating

that the above assignment is optimal, they claim that the

solution is not unique and another optimal assignment

can be obtained as A, B, C, D are assign to II, IV, III, I

respectively and minimum assignment cost is [12 , 20].

Hence their result and their conclusion violate the

concept of optimality.

Now we shall solve the same interval assignment

problem given in table 3 by applying the method

proposed in this paper.

Let us express all the interval parameters 1 2a = [a , a ] in

terms of midpoint and width as 1 2a [a ,a ] m(a),w(a) .

Now the given interval transportation problem becomes


I II III IV

A <10,1> <5,1> <13,1> <15,1> B <3,1> <9,1> <18,1> <3,1> C <10,1> <7,1> <3,1> <2,1> D <5,1> <1,1> <9,1> <7,1>


I II III IV

A <5,1> <0,1> <8,1> <10,1> B <0,1> <6,1> <15,1> <0,1>

C <8,1> <5,1> <1,1> <0,1> D <4,1> <0,1> <8,1> <6,1>


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I II III IV

A <5,1> <0,1> <7,1> <10,1> B <0,1> <6,1> <14,1> <0,1> C <8,1> <5,1> <0,1> <0,1> D <4,1> <0,1> <7,1> <6,1>


I II III IV

A <5,1> <0,1> <7,1> <10,1> B <0,1> <6,1> <14,1> <0,1> C <8,1> <5,1> <0,1> <0,1>

D <4,1> <0,1> <7,1> <6,1>


I II III IV

A <1,1> <0,1> <3,1> <6,1> B <0,1> <10,1> <14,1> <0,1>

C <8,1> <9,1> <0,1> <0,1> D <0,1> <0,1> <3,1> <2,1>


I II III IV

A <10,1> <5,1> <13,1> <15,1>

B <3,1> <9,1> <18,1> <3,1>

C <10,1> <7,1> <3,1> <2,1>

D <5,1> <1,1> <9,1> <7,1>

The optimal assignment schedule is given by

A II, B IV, C III, D I

The optimal assignment cost

= <5, 1> + <3, 1> + <3, 1> + <5, 1>

= <16, 1>

= [15, 17]

It is to be noted that our solution [15, 17] is very much

sharper than the solution [12, 20] obtained by Sarangam

Majumdar

Example 4.2

Let us consider an interval assignment problem with rows representing 4 areas A, B, C, D and columns representing the salesmen I, II,III, IV. The cost matrix

ij(C ) is given whose elements are interval numbers. The

problem is to find the optimal assignment so that the total cost of area assignment becomes minimum.


I II III IV

A [2,4] [4,6] [6,8] [0,2]

B [8,10] [7,9] [11,13] [9,11]

C [12,14] [7,8] [13,15] [1,3]

D [4,6] [6,8] [9,11] [5,7]

Let us express all the interval parameters 1 2a = [a , a ] in

terms of midpoint and width as 1 2a [a ,a ] m(a),w(a) .

Now the given interval transportation problem becomes


I II III IV

A <3,1> <5,1> <7,1> <1,1>

B <9,1> <8,1> <12,1> <10,1>

C <13,1> <8,1> <14,1> <2,1>

D <5,1> <7,1> <10,1> <6,1>


I II III IV

A <2,1> <4,1> <6,1> <0,1> B <1,1> <0,1> <4,1> <2,1>

C <11,1> <6,1> <12,1> <0,1> D <0,1> <2,1> <5,1> <1,1>


I II III IV

A <2,1> <4,1> <2,1> <0,1> B <1,1> <0,1> <0,1> <2,1> C <11,1> <6,1> <8,1> <0,1> D <0,1> <2,1> <1,1> <1,1>


I II III IV

A <2,1> <4,1> <2,1> <0,1> B <1,1> <0,1> <0,1> <2,1> C <11,1> <6,1> <8,1> <0,1> D <0,1> <2,1> <1,1> <1,1>


I II III IV

A <0,1> <2,1> <0,1> <0,1>

B <1,1> <0,1> <0,1> <4,1>

C <9,1> <4,1> <6,1> <0,1>

D <0,1> <2,1> <1,1> <3,1>

The optimal assignment schedule is given by

A III, B II, C IV, D I

The optimal assignment cost

= <7, 1> + <8, 1> + <2, 1> + <5, 1>

= <22, 1>

= [21, 23]

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5. CONCLUSION In this paper, a new approach to solve the assignment

problems with generalized interval arithmetic is

proposed. To show the efficacy of the proposed method

numerical examples are solved and corresponding results

are compared.

REFERENCES

[1]. Albrecher, H. (2005) “A note on the asymptotic behaviour of bottleneck problems,” Operations Research Letters, 33, 183–186.

[2]. Atanu Sengupta and Tapan Kumar Pal, Theory and Methodology: On comparing interval numbers, European Journal of Operational Research, 27 (2000), 28 - 43.

[3]. Aldous, D. (1992) “Asymptotics in the random assignment problem,” Probability Theory and Related Fields, 93, 507–534.

[4]. Alvis, D. and Lai, C.W. (1988) “The probabilistic analysis of a heuristic for the assignment problem,” SIAM Journal on Computing, 17, 732–741.

[5]. Anstreicher, K. M. (2003) “Recent advances in the solution of quadratic assignment problems,” Mathematical Programming, 97 (1–2), 27–42.

[6]. Buck, M.W., Chan, C. S., and Robbins, D. P. (2002) “On the Expected Value of the Minimum Assignment,” Random Structures & Algorithms, 21 (1), 33–58.

[7]. Burkard, R. E. (1985) “Time-slot assignment for TDMA systems,” Computing, 35 (2), 99–112.

[8]. Burkard, R. E. and Fincke, U. (1982a) “The Asymptotic Probabilistic Behavior of Quadratic Sum Assignment Problems,” Zeitschrift für Operations Research, 27, 73–81.

[9]. Burkard, R. E. and Fincke, U. (1982b) “On random quadratic bottleneck assignment problems,” Mathematical Programming, 23, 227–232.

[10]. Burkard, R. E., Hahn, W., and Zimmermann, U. (1977) “An algebraic approach to assignment problems,” Mathematical Programming, 12, 318–327.

[11]. Cloud, Michael J.; Moore, Ramon E.; Kearfott, R. Baker, Introduction to Interval Analysis. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-669-1(2009).

[12]. Gabow, H. N. and Tarjan, R. E. (1988) “Algorithms for two bottleneck assignment problems,” Journal of Algorithms, 9 (3), 411–417 .

[13]. Hamdy A. Taha, Operations Research (Eighth Edition)(2008), Pearson, ISBN -9780131889231.

[14]. Kuhn, H. W. (1955) “The Hungarian method for the assignment problem,” Naval Research Logistics Quarterly, 2 (1–2), 83–87.

[15]. Krokhmal, P., Grundel, D., and Pardalos, P. (2007) “Asymptotic Behavior of the Expected Optimal Value of the Multidimensional Assignment Problem,” Mathematical Programming, 109 (2–3), 525–551.

[16]. Kellerer, H. and Wirsching, G. (1998) “Bottleneck quadratic assignment problems and the bandwidth problem,” Asia-Pacific Journal of Operational Research, 15 (2), 169–177.

[17]. Li, Y. and Pardalos, P. M. (1992) “Generating Quadratic Assignment Test Problems with Known Optimal Permutations,” Computational Optimization and Applications, 1 (2), 163–184.

[18]. Merz, P. and Freisleben, B. (2000) “Fitness landscape analysis and memetic algorithms for the quadratic assignment problem,” IEEE Transactions on Evolutionary Computation, 4 (4), 337–352.

[19]. Ming Ma, Menahem Friedman and Abraham Kandel.1999. A new fuzzy arithmetic. Fuzzy sets and systems. 108: 83-90.

[20]. Nair, C. (2005) Proofs of the Parisi and Coppersmith-Sorkin conjectures in the random assignment problem, Ph.D. thesis, Stanford University.

[21]. Olin, B. (1992) Asymptotic properties of the random assignment problem, PhD thesis, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden.

[22]. Pardalos, P. M. and Pitsoulis, L. (Eds.) (2000) Nonlinear Assignment Problems: Algorithms and Applications, Kluwer Academic Publishers.

[23]. Sarangam Majumdar, Interval Linear Assignment Problems, Universal Journal of Applied Mathematics, 1(1): 14 -16, 2013.

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ASSIGNMENT PROBLEM WITH GENERALIZED INTERVAL ARITHMETIC · ASSIGNMENT PROBLEM WITH GENERALIZED INTERVAL ARITHMETIC . G. Ramesh. and. K. Ganesan. Abstract - Linear assignment problems